Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,14)(8,11)(9,13)(10,15)(12,17)(16,19)(18,20);(5,7)(6,10,13,18)(8,15)(9,16)(11,19,12,14)(17,20);(10,18)(14,19);(6,11,13,12)(10,19,18,14)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72258084, #72260535, #72261017, … |
| maximal normal subgroup of | #72258084, #72260535, #72261017, … |
| maximal quotient of | #72234288 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271117);